Let’s address the first question you have first.
I think one of the most helpful things to do on this scenario is the “Zen trick”. Do you remember that? If not, review the second video in Unit 5 of Basic Electricity.
Do the Zen trick on the ignitor. What do you find?
Then do it on the booster. What do you find?
Can your right and left hands reach out to L1 and N without going through any other loads?